The novel point of this analysis is a direct use of the Reggeon exchange technique for the description of the reactions πN → two mesons + N at large energies of the initial pion. This approach allows us to describe simultaneously distributions over M (invariant mass of two mesons) and t (momentum transfer squared to nucleons). Making use of this technique, the following resonances (as well as corresponding bare states), produced in the πN→π0π0N reaction, are studied: f0(980), f0(1300) [f0(1370) in PDG notation], f0(1200–1600), f0(1500), f0(1750), f2(1270), f2(1525), f2(1565), f2(2020), f4(2025). Adding data for the reactions [Formula: see text] (at rest, from liquid H2) →π0π0π0, π0π0η, π0ηη and [Formula: see text] (at rest, from gaseous H2) →π0π0π0, π0π0η, π0ηη, we have performed a simultaneous K matrix fit of two-meson spectra in all these reactions. The results of combined fits to the above-listed isoscalar fJ states and to isovector ones, a0(980), a0(1475), a2(1320), are presented.